It is shown that a trivial version of polarization is sufficient to produce separating systems of polynomial invariants: if two points in the direct sum of the G–modules W and m copies of V can be separated by polynomial invariants, then they can be separated by invariants depending only on ≤ 2 dim(V ) variables of type V ; when G is reductive, invariants depending only on ≤ dim(V ) + 1 variabl...

Numerical procedures are proposed for triangularizing polynomial matrices over the eld of polynomial fractions and over the ring of polynomials. They are based on two standard polynomial techniques: Sylvester matrices and interpolation. In contrast to other triangularization methods, the algorithms described in this paper only rely on well-worked numerically reliable tools. They can also be use...

Index 74 Preface The three subjects of the title (codes, matroids, and permutation groups) have many interconnections. In particular, in each case, there is a polynomial which captures a lot of information about the structure: we have the weight enumerator of a code, the Tutte polynomial (or rank polynomial) of a matroid, and the cycle index of a permutation group. With any code is associated a...

Interval Newton methods are widely used to find reliable enclosures for the roots of the polynomial systems. But the computation process needs the evaluation of an enclosure for the derivative of the polynomial and evaluation of the polynomial at a particular point. Again, the subdivisions (if any are needed) will require repeated evaluation of the function values. We propose an alternative app...

We give the first polynomial-time algorithm for checking whether the Galois group Gal (f) of an input polynomial f(X) ∈ Q[X] is nilpotent: the running time of our algorithm is bounded by a polynomial in the size of the coefficients of f and the degree of f . Additionally, we give a deterministic polynomial-time algorithm that, when given as input a polynomial f(X) ∈ Q[X] with nilpotent Galois g...